Problem: A white cylindrical silo has a diameter of 30 feet and a height of 80 feet.  A red stripe with a horizontal  width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?

[asy]
size(250);defaultpen(linewidth(0.8));
draw(ellipse(origin, 3, 1));
fill((3,0)--(3,2)--(-3,2)--(-3,0)--cycle, white);
draw((3,0)--(3,16)^^(-3,0)--(-3,16));
draw((0, 15)--(3, 12)^^(0, 16)--(3, 13));
filldraw(ellipse((0, 16), 3, 1), white, black);
draw((-3,11)--(3, 5)^^(-3,10)--(3, 4));
draw((-3,2)--(0,-1)^^(-3,1)--(-1,-0.89));
draw((0,-1)--(0,15), dashed);
draw((3,-2)--(3,-4)^^(-3,-2)--(-3,-4));
draw((-7,0)--(-5,0)^^(-7,16)--(-5,16));
draw((3,-3)--(-3,-3), Arrows(6));
draw((-6,0)--(-6,16), Arrows(6));
draw((-2,9)--(-1,9), Arrows(3));
label("$3$", (-1.375,9.05), dir(260), UnFill);
label("$A$", (0,15), N);
label("$B$", (0,-1), NE);
label("$30$", (0, -3), S);
label("$80$", (-6, 8), W);
[/asy]
Solution: If the stripe were cut from the silo and spread flat, it would form a parallelogram 3 feet wide and 80 feet high. So the area of the stripe is $3(80)=\boxed{240}$ square feet.

Notice that neither the diameter of the cylinder nor the number of times the stripe wrapped around the cylinder factored into our computation for the area of the stripe. At first, this may sound counter-intuitive. An area of 240 square feet is what we would expect for a perfectly rectangular stripe that went straight up the side of the cylinder.

However, note that no matter how many times the stripe is wrapped around the cylinder, its base and height (which are perpendicular) are always preserved. So, the area remains the same. Consider the following stripes which have been "unwound" from a cylinder with height 80 feet.

[asy]
size(400);
real s=8;
pair A=(0,0), B=(1.5,0), C=(1.5,20), D=(0,20);
draw(A--B--C--D--cycle);
label("$3$", (C+D)/2, N);
label("$80$", (A+D)/2, W);

draw(shift(s)*(shift(20)*A--shift(20)*B--C--D--cycle));
label("$3$", shift(s)*((C+D)/2), N);
draw(shift(s)*((0,0)--D), dashed);
label("$80$", shift(s)*(((0,0)+D)/2), W);

draw(shift(4.5s)*(shift(40)*A--shift(40)*B--C--D--cycle));
label("$3$", shift(4.5s)*((C+D)/2), N);
draw(shift(4.5s)*((0,0)--D), dashed);
label("$80$", shift(4.5s)*(((0,0)+D)/2), W);
[/asy]

Regardless of how many times the stripes were wrapped around the cylinder, each stripe has base 3 feet and height 80 feet, giving area 240 sq ft.